Nonzero Steklov Eigenvalue Research Articles

An upper bound for the first nonzero Steklov eigenvalue

Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds $\Sect_g\leq \kappa$ for $\kappa\leq 0$ and $\Ric_g\geq(n-1)Kg$ for $K\leq 0$. We prove that for any bounded domain $\Omega \subset M^n$ with diameter $d$ and Lipschitz boundary, if $\Omega^*$ is a geodesic ball in the simply connected space form with constant sectional curvature $\kappa$ enclosing the same volume as $\Omega$, then $\sigma_1(\Omega) \leq C \sigma_1(\Omega^*)$, where $\sigma_1(\Omega)$ and $ \sigma_1(\Omega^*)$ denote the first nonzero Steklov eigenvalues of $\Omega$ and $\Omega^*$ respectively, and $C=C(n,\kappa, K, d)$ is an explicit constant. When $\kappa=K$, we have $C=1$ and recover the Brock-Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.

Estimates for low Steklov eigenvalues of surfaces with several boundary components

R\'esum\'eIn this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $$\sigma _1$$ σ 1 of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

Upper bounds for the Steklov eigenvalues on trees

In this paper, we study the upper bounds for discrete Steklov eigenvalues on trees via geometric quantities. For a finite tree, we prove sharp upper bounds for the first nonzero Steklov eigenvalue by the reciprocal of the size of the boundary and the diameter respectively. We also prove similar estimates for higher order Steklov eigenvalues.

The overdetermined problem and lower bound estimate on the first nonzero Steklov eigenvalue of p-Laplacian

The overdetermined problem and lower bound estimate on the first nonzero Steklov eigenvalue of p-Laplacian

Where to place a spherical obstacle so as to maximize the first nonzero Steklov eigenvalue

We prove that among all doubly connected domains of ℝnof the formB1\B̅2, whereB1andB2are open balls of fixed radii such thatB̅2⊂B1, the first nonzero Steklov eigenvalue achieves its maximal value uniquely when the balls are concentric. Furthermore, we show that the ideas of our proof also apply to a mixed boundary conditions eigenvalue problem found in literature.

Large Steklov eigenvalues via homogenisation on manifolds

Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of 8pi for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than 2pi . This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus zero in the unit ball with even larger area. We prove that the first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they are consistent with a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois–El Soufi–Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.

Weinstock inequality in higher dimensions

We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in $\mathbb{R}^n$, for $n \geq 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultaneously the surface area, the volume and the boundary momentum of convex sets. As a by-product, we also obtain some isoperimetric inequalities for the first Wentzell eigenvalue.

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj(Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σj(Σ, g) is the jth nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components.

Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems

We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent results of Han and Hua, who obtained a similar result in the case of mathbb {Z}^n. We obtain the result using metric properties of Cayley graphs associated to groups of polynomial growth.

Higher order Cheeger inequalities for Steklov eigenvalues

We prove a lower bound for the k-th Steklov eigenvalues in terms of an isoperimetric constant called the k-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considering a family of accelerated Markov operators in the finite and measurable situations and of mass concentration deformations of the Laplace-Beltrami operator in the manifold setting which converges uniformly to the Steklov operator. As an intermediary step in the proof of the higher order Cheeger type inequality, we define the Dirichlet-Steklov connectivity spectrum and show that the Dirichlet connectivity spectra of this family of operators converges to (or is bounded by) the Dirichlet-Steklov spectrum uniformly. Moreover, we obtain bounds for the Steklov eigenvalues in terms of its Dirichlet-Steklov connectivity spectrum which is interesting in its own right and is more robust than the higher order Cheeger type inequalities. The Dirichlet-Steklov spectrum is closely related to the Cheeger-Steklov constants. Resume Pour tout k P N, une borne inferieure pour la k-ieme valeur propre de Steklov en termes d'une constante isoperimetrique, appelee la k-ieme constante de Cheeger-Steklov, est obtenue dans trois situations differentes : espaces finis, espaces mesurables et varietes riemanniennes. Ces bornes inferieures peuvent etre considerees comme des inegalites de type Cheeger d'ordre superieur pour les valeurs propres de Steklov. En particulier, elles etendent l'inegalite de type Cheeger pour la premi ere valeur propre non nulle de Steklov etudiee par Escobar en 1997 et par Jammes en 2015. La technique developpee pour obtenir ces bornes inferieure utilise une famille d'operateurs de Markov acceleres dans les situations finies et mesurables et une famille d'operateurs de Laplace-Beltrami deformes et concentres pres de lafronti ere. Lors d'une etape intermediaire de la preuve de l'inegalite de type Cheeger d'ordre superieur, nous definissons le spectre de connectivite de Dirichlet-Steklov et nous montrons que les spectres de connectivite de Dirichlet de cette famille d'operateurs convergent uniformement vers (ou sont bornes par) le spectre de Dirichlet-Steklov. De plus, nous obtenons des bornes pour les valeurs propres de Steklov en termes du spectre de connectivite de Dirichlet-Steklov, ce dernier etant interessant en lui-meme. Il est aussi plus robuste que les inegalites de type Cheeger d'ordre superieur. Le spectre de Dirichlet-Steklov et les constantes de Cheeger-Steklov sont etroitement lies.

Eigenvalue comparisons in Steklov eigenvalue problem and some other eigenvalue estimates

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted Laplacian, which can be seen as a natural generalization of the classical Steklov eigenvalue problem, have been obtained.

Evolution of the Steklov eigenvalue under geodesic curvature flow

On a two-dimensional compact Riemannian manifold with boundary, we prove that the first nonzero Steklov eigenvalue is nondecreasing along the unnormalized geodesic curvature flow if the initial metric has positive geodesic curvature and vanishing Gaussian curvature. Using the normalized geodesic curvature flow, we also obtain some estimate for the first nonzero Steklov eigenvalue. On the other hand, we prove that the compact soliton of the geodesic curvature flow must be the trivial one.

Shape optimization for the Steklov problem in higher dimensions

We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in Rn when n≥3. This is in contrast to the situation when n=2, where a result of Weinstock from 1954 shows that the disk uniquely maximizes the first Steklov eigenvalue among all simply connected domains in the plane having the same boundary length. When n≥3, we show that increasing the number of boundary components does not increase the normalized (by boundary volume) first Steklov eigenvalue. This is in contrast to recent results which have been obtained for surfaces and for convex domains.

Lower bounds for the first eigenvalue of the Steklov problem on graphs

We give lower bounds for the first non-zero Steklov eigenvalue on connected graphs. These bounds depend on the extrinsic diameter of the boundary and not on the diameter of the graph. We obtain a lower bound which is sharp when the cardinal of the boundary is 2, and asymptotically sharp as the diameter of the boundary tends to infinity in the other cases. We also investigate the case of weighted graphs and compare our result to the Cheeger inequality.

From Steklov to Neumann and Beyond, via Robin: The Szegő Way

AbstractThe second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form$\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$, and$\unicode[STIX]{x1D6FC}$lies between$-2\unicode[STIX]{x1D70B}$and$2\unicode[STIX]{x1D70B}$. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

A characterization of the disk by eigenfunction of the Steklov eigenvalue

Suppose M is homeomorphic to 2-dimensional sphere and let \(f_i\), \(1\le i\le 3\), be the first eigenfunctions of the Laplacian on M. Cheng proved (Proc Am Math Soc 55(2):379–381, 1976) that if \(\sum _{i=1}^3 f_i^2\) is a constant, then M is isometric to a sphere of constant curvature. In view of the similarities between eigenvalues of the Laplacian and Steklov eigenvalue, we study eigenfuction of the first nonzero Steklov eigenvalue of a 2-dimensional compact manifold with boundary \(\partial M\). Suppose M is a domain equipped with the flat metric g, and let f be an eigenfunction of the first nonzero Steklov’s eigenvalue. We prove that if \(\nabla _g f\) is parallel along \(\partial M\), the Lie bracket of the vector field orthogonal to \(\nabla _g f\) and the tangent to \(\partial M\) is zero, and \(|\nabla _g f|^2\) is constant in M, then (M, g) is isometric to the disk equipped with the flat metric. We also prove that if the eigenfunctions \(f_1, f_2\) corresponding to the first nonzero Steklov eigenvalue satisfy that \(f_1^2+f_2^2\) is constant on \(\partial M\) and \(\varphi _*(g)\) is a Riemannian metric conformal to the flat metric of \(D^2\) where \(\varphi =(f_1,f_2)\), then (M, g) is isometric to the disk equipped with the flat metric. In another direction, we prove that a simply connected domain in the hyperbolic space \(\mathbb {H}^3\) such that its Steklov eigenvalues are the same as a geodesic ball must be isometric to the geodesic ball.

Some sharp Hodge Laplacian and Steklov eigenvalue estimates for differential forms

We give some sharp lower bounds of the first eigenvalue for the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold. We also give some sharp estimates for the first nonzero Steklov eigenvalue for differential forms.

Estimates of the first Steklov eigenvalue of properly embedded minimal hypersurfaces with free boundary

We consider a properly embedded minimal hypersurfacewith free boundary in a compact n-dimensional Riemannian manifold M be with nonnegative Ricci curvature and strictly convex boundary. Here, we obtain a new estimate from below for the first nonzero Steklov eigenvalue.

Sharp eigenvalue bounds and minimal surfaces in the ball

We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball B^n for some n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid, the unique free boundary surface of revolution in B^3. We also show that the unique solution on the Mobius band is achieved by an explicit S^1 invariant embedding in B^4 as a free boundary surface, the critical Mobius band. For oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in B^3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4pi. We also prove multiplicity bounds on sigma_1 in terms of the topology, and we give a lower bound on the Morse index for the area functional for free boundary surfaces in the ball.

The spectral gap of graphs and Steklov eigenvalues on surfaces

Using expander graphs, we construct a sequence $\{\Omega_N\}_{N\in\mathbb{N}}$ of smooth compact surfaces with boundary of perimeter $N$, and with the first non-zero Steklov eigenvalue $\sigma_1(\Omega_N)$ uniformly bounded away from zero. This answers a question which was raised in [10]. The sequence $\sigma_1(\Omega_N) L(\partial\Omega_n)$ grows linearly with the genus of $\Omega_N$, which is the optimal growth rate.